b Graph the data points and analyze their behavior.
C
c Use your calculator to perform the corresponding regression.
D
d Use the equation found in Part C.
E
e Consider the context of the real world being modeled.
A
aAverage Rate of Change 1955-1965: - 0.48 % per year Average Rate of Change 1955-1965: - 0.75 % per year
B
bGraph:
The data suggests using a linear model.
C
c y = - 0.4464 x + 57.77
D
d The prediction for the percentage for 2020 is approximately 4.2 %.
E
e See solution.
Practice makes perfect
a We are given the data table shown below, showing the percentage of the U.S. labor force in unions for selected years between 1955 and 2005.
Year
%
1955
33.2
1960
31.4
1965
28.4
1970
27.3
1975
25.5
1980
21.9
1985
18
1990
16.1
1995
14.9
2000
13.5
2005
12.5
Recall that the average rate of change between two data points is equal to the slope between them, as this is just the quotient between the changes of the dependent and the independent variable at those points. Therefore, we can use the Slope Formula.
m = y_2-y_1/x_2-x_1
In the equation above, (x_1,y_1) and (x_2,y_2) represent the two known points. We can now substitute the points of interest. Let's start by finding the average rate of change between 1955 and 1965.
Therefore, the average rate of change between 1900 and 1965 is - 0.48 % per year, while between 1975 and 1985 it was - 0.75 % per year.
b We can graph the points by taking into account the years after 1900 in the x-axis and the corresponding percentage in the y-axis.
As we can see from the graph above, the data set suggests that a linear model is most appropriate.
c To perform the linear regression from the data values, we enter them into the calculator's lists. We can do this by pressing STAT, and then choosing Edit.
The lists will shown after this, and we will be able to input the percentages in L1 and the years in L2.
To view the linear regression analysis of the data set, we press STAT, move to the right to view the CALC options, and then choose the option LinReg from the list.
We can round the values for the parameters to a=- 0.4464 x and b=57.77. With this information, we can write the equation for our linear model.
y = - 0.4464 x + 57.77
d To find the percent of the labor force in unions in the year 2020, we can evaluate the equation found in Part C at x=120.
We can see that the prediction for the percentage for 2020 is around 4.2 %.
e No, even if the coefficient of determination R^2 is close to 1, which indicates a good fit for the model, as the linear model has a negative slope, it predicts that the percentage will eventually be 0 and then become negative. This does not make sense for the context of the real world situation being modeled.
